There is ample reason for caution because Mr. Trump has a history of making attention-grabbing and politically explosive claims that have no basis in fact. As things stand, a plausible explanation is that Mr. Trump was merely riffing off a March 3 article on the alt-right website Breitbart.com. It laid out a theory circulating in some conservative circles that President Barack Obama sought to sabotage Mr. Trump through surveillance.

The episode has heightened interest in several related surveillance issues.

Can a president order the wiretapping of an American?

Not legally. There are two lawful ways that allow government officials to target people in the United States for surveillance and to collect the contents of their phone calls and emails: criminal wiretaps, called “Title III” warrants, and national security wiretaps, called Foreign Intelligence Surveillance Act, or FISA, orders. Both types of wiretapping are ordered by federal judges, after applications from the Justice Department.

If there was a wiretap order targeting Mr. Trump or his associates, what would that mean?

If it was a criminal wiretap, it would mean that the Justice Department had gathered sufficient evidence to convince a federal judge that someone using the phone number or email address probably committed a serious crime. If it was a national security wiretap, it would mean a federal judge on the Foreign Intelligence Surveillance Court had a basis to believe the target was probably an agent of a foreign power, like Russia.

Could a judge have approved tapping something at Trump Tower for another reason?

Yes. For example, FISA orders have two parts. After determining that there is probable cause to believe that the target is a foreign agent, a judge also has to approve directing surveillance at a particular “facility,” like a phone line or an email address, that the target is probably using to communicate. So in theory, if there was reason to believe that some other lawful target was communicating from Trump Tower, a judge could have authorized surveillance at a facility there for that reason.

What about the computer server registered to Trump Tower?

Several news media outlets have reported that investigators last year were puzzled by data transmissions between a computer server registered to Trump Tower and a computer server associated with a Russian bank. Although Mr. Trump on Twitter talked about his “phones,” in theory a judge might determine that the computer address of the server was a facility being used by a foreign power, Russia, to communicate, and authorize surveillance of it.

Isn’t there a report about an October surveillance court order involving that server?

Yes. The Breitbart story relied heavily on a Nov. 7 article by a British writer on the news and opinion website HeatStreet. It claimed that the Foreign Intelligence Surveillance Court had approved a “warrant” in October in connection with activity between Russian banks and the server, which it said — and Breitbart repeated — was in Trump Tower. It further stated that “it is thought in the intelligence community” that this purported warrant permitted the collection of emails and other communications of Americans connected to the server investigation, which “thus covers Donald Trump.”

As things stand, there are reasons to be skeptical. HeatStreet had vague sourcing — two “sources with links to the counterintelligence community” — and it does not regularly publish investigative stories about American intelligence or law enforcement operations. To date, reporters for The New York Times with demonstrated sources in that world have been unable to corroborate that the court issued any such order. (Computer specialists have also pointed out that the server in question does not appear to be located in Trump Tower.) On Sunday, James Clapper, who was the director of national intelligence until Jan. 20, denied to NBC News that the Foreign Intelligence Surveillance Court had approved monitoring at Trump Tower.

What about wiretapping Michael Flynn’s calls with the Russian ambassador?

In a related matter, some conservatives have raised alarms about why the government eavesdropped on the late-December phone calls about sanctions between the Russian ambassador and Michael Flynn, who resigned last month as Mr. Trump’s national security adviser. But national security specialists say it is routine that counterintelligence officials would monitor the Russian ambassador, who is a clearly an agent of a foreign power, and so would “incidentally” pick up Mr. Flynn, too. The rules generally require officials to “minimize” the privacy intrusion by masking the names and data of incidentally intercepted Americans before sharing reports or transcripts of those calls more widely within the government. However, there is an exception if the conversation constituted foreign intelligence and the American’s identity is necessary to understand its significance, as would be the case with Mr. Flynn’s discussion of sanctions.

Did Obama loosen rules for sharing surveillance?

Yes. In January, the Obama administration changed the rules governing how the National Security Agency may share raw surveillance it intercepts abroad, where the Foreign Intelligence Surveillance Act does not apply and the agency may vacuum up private communications in bulk. Previously, only analysts at the agency were permitted to search through the raw trove of this information, applying “minimization” rules to information about Americans before sharing anything more widely in the government. Now, analysts from the other 16 intelligence agencies — including the F.B.I. — can sift through certain streams of raw intercepts themselves, applying “minimization” rules afterward.

Some conservative commentators, like Fox News’s Sean Hannity, have floated the theory that news media leaks about contacts between people affiliated with Mr. Trump and Russians may be attributable to this change, asking why Mr. Obama opened the door to many more officials across the government having access to raw National Security Agency intercepts. Several factors complicate that insinuation. The change did not affect collection on American soil under the Foreign Intelligence Surveillance Act, which several agencies were already able to share in raw form. Moreover, the recent change had been in the works for years. President George W. Bush set it in motion in 2008, and the bureaucratic deliberations were well developed long before it became clear that Mr. Trump would become president.

Correction:

An earlier version of this article incorrectly referred to HeatStreet as a “British blog.” The HeatStreet article in question was written by HeatStreet’s then-editor, who is British, but the website is based in New York. This article has also been updated to clarify that a server that attracted investigators’ suspicions is registered to Trump Tower but appears not to be located there.

A version of this article appears in print on , on Page A13 of the New York edition with the headline: What Can Be Gleaned From the President’s Allegations on Twitter of Wiretapping. Order Reprints | Today’s Paper | Subscribe